Jeffrey P. Thomas

Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique

A harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented. The analysis exploits the fact that many unsteady e ows of interest in turbomachinery are periodic in time. Thus, the unsteady e ow conservation variables may be represented by a Fourier series in time with spatially varying coefe cients. This assumption leads to a harmonic balance form of the Euler or Navier ‐Stokes equations, which, in turn, can be solved efe ciently as a steady problem using conventional computational e uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efe cient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous e ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear e ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the e ow. Furthermore, in some cases, e uid nonlinearities are found to be important for surprisingly small blade vibrations.

Proper Orthogonal Decomposition Technique for Transonic Unsteady Aerodynamic Flows

A new method for constructing reduced-order models (ROM) of unsteady small-disturbance flows is presented. The reduced-order models are constructed using basis vectors determined from the proper orthogonal decomposition (POD) of an ensemble of small-disturbance frequency-domain solutions. Each of the individual frequency-domain solutions is computed using an efficient time-linearized flow solver. We show that reduced-order models can be constructed using just a handful of POD basis vectors, producing low-order but highly accurate models of the unsteady flow over a wide range of frequencies. We apply the POD/ROM technique to compute the unsteady aerodynamic and aeroelastic behavior of an isolated transonic airfoil and to a two-dimensional cascade of airfoils

Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter and Limit Cycle Oscillations

By the use of a state-of-the-art computational e uid dynamic (CFD) method to model nonlinear steady and unsteady transonice owsin conjunction with a linearstructural model,an investigationismadeinto how nonlinear aerodynamics can effect the divergence, e utter, and limit-cycle oscillation (LCO) characteristics of a transonic airfoil cone guration. A single-degree-of-freedom (DOF) model is studied for divergence, and one- and two-DOF models are studied for e utter and LCO. A harmonicbalancemethod in conjunction with the CFD solver is used to determine the aerodynamics for e nite amplitude unsteady excitations of a prescribed frequency. A procedure for determining the LCO solution is also presented. For the cone guration investigated, nonlinear aerodynamic effects are found to produce a favorable transonic divergence trend and unstable and stable LCO solutions, respectively, for the one- and two-DOF e utter models. Nomenclature a = nondimensional location of airfoil elastic axis, e=b b, c = semichord and chord, respectively cl, cm = coefe cients of lift and moment about elastic axis, respectively e = location of airfoil elastic axis, measured positive aft of airfoil midchord h, ® = airfoil plunge and pitch degrees of freedom I® = second moment of inertia of airfoil about elastic axis

Three-Dimensional Transonic Aeroelasticity Using Proper Orthogonal Decomposition-Based Reduced-Order Models

The proper orthogonal decomposition (POD) based reduced order modeling (ROM) technique for modeling unsteady frequency domain aerodynamics is developed for a large scale computational model of an inviscid flow transonic wing configuration. Using the methodology, it is shown that a computational fluid dynamic (CFD) model with over a three quarters of a million degrees of freedom can be reduced to a system with just a few dozen degrees of freedom, while still retaining the accuracy of the unsteady aerodynamics of the full system representation. Furthermore, POD vectors generated from unsteady flow solution snapshots based on one set of structural mode shapes can be used for different structural mode shapes so long as solution snapshots at the endpoints of the frequency range of interest are included in the overall snapshot ensemble. Thus, the snapshot computation aspect of the method, which is the most computationally expensive part of the procedure, does not have to be fully repeated as different structural configurations are considered.

Modeling Viscous Transonic Limit Cycle Oscillation Behavior Using a Harmonic Balance Approach

Presented is a harmonic-balance computational fluid dynamic approach for modeling limit-cycle oscillation behavior of aeroelastic airfoil configurations in a viscous transonic flow. For the NLR 7301 airfoil configuration studied, accounting for viscous effects is shown to significantly influence computed limit-cycle oscillation trends when compared to an inviscid analysis. A methodology for accounting for changes in mean angle of attack during limit-cycle oscillation is also developed.

Reduced-order modelling of unsteady small-disturbance flows using a frequency-domain proper orthogonal decomposition technique

A new method for constructing reduced-order models (ROM) of unsteady small-disturbance flows is presented. The reduced-order models are constructed using basis vectors determined from the proper orthogonal decomposition (POD) of an ensemble of small-disturbance frequencydomain solutions. Each of the individual frequency-domain solutions is computed using an efficient time-linearized flow solver. We show that reduced-order models can be constructed using just a handful of POD basis vectors, producing low-order but highly accurate models of the unsteady flow over a wide range of frequencies. In this paper, we apply the POD/ROM technique to compute the unsteady aerodynamic and aeroelastic behavior of an isolated nansonic airfoil, and to a two-dimensional cascade of airfoils. Nomenclature A = matrix defining homogeneous part of discretized aerodynamic operator A = reduced-order form of A. b = airfoil semi-chord b = vector defining inhomogeneous part of discretized aerodynamic operator Ba, B1 = matrices relating airfoil motion h and h to b c c

A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator

The present study focuses on a novel harmonic balance formulation, which is much easier to implement than the standard/classical harmonic balance method for complex nonlinear mathematical models and algorithms. Both harmonic balance approaches are applied to Duffings oscillator to demonstrate the advantages and disadvantages of the two approaches. A fundamental understanding of the difference between these two methods is achieved, and the properties of each method are analyzed in detail.

Discrete Adjoint Approach for Modeling Unsteady Aerodynamic Design Sensitivities

A discrete adjoint approach is presented for computing steady and unsteady aerodynamic design sensitivities for compressible viscous flows about airfoil configurations. The nominal flow solver method is based on a harmonic balance solution technique, which is capable of modeling both steady and nonlinear periodic unsteady flows. The computer code for the discrete adjoint solver, which is derived from the nominal harmonic balance solver, has been generated with the aid of the advanced automatic differentiation software tool known as TAF (Transformation of Algorithms in FORTRAN).

Using Automatic Differentiation to Create a Nonlinear Reduced-Order-Model Aerodynamic Solver

A novel nonlinear reduced-order-modeling technique for computational aerodynamics and aeroelasticity is presented. The method is based on a Taylor series expansion of a frequency-domain harmonic balance computational fluid dynamicsolver residual.The first- andsecond-order gradientmatrices andtensorsof the Taylorseries expansion are computed using automatic differentiation via FORTRAN 90=95 operator overloading. A Ritz-type expansion using proper orthogonal decomposition shapes is then used in the Taylor series expansion to create the nonlinearreduced-order model. The nonlinear reduced-order-modeling technique is applied to a viscous flow about anaeroelastic NLR 7301 airfoil modelto determine limit cycle oscillations. Computational times are decreased from hours to seconds using the nonlinear reduced-order model.

Unsteady Flow Computation Using a Harmonic Balance Approach Implemented about the OVERFLOW 2 Flow Solver

A novel approach for implementing a nonlinear unsteady frequency domain harmonic balance solution technique about existing implicit computational fluid dynamic flow solvers is presented. This approach uses an explicit discretization of the harmonic balance source term, which enables the harmonic balance method to be applied to existing implicit flow solvers with minimal need for modification to the underlying implicit flow solver code. The resulting harmonic balance solver can then be used for modeling nonlinear periodic unsteady flows. The methodology is applied to the OVERFLOW 2 flow solver code, and results are presented for transonic viscous flow past an unsteady pitching airfoil section. Unsteady aerodynamic and aeroelastic results for the F-16 fighter wing are also presented.